[Paper Review] Randomly coupled differential equations with correlations

László Erdős, Torben Krüger+1 more|arXiv (Cornell University)|Aug 14, 2019

Quantum many-body systems14 references4 citations

TL;DR

This paper analyzes the long-time behavior of large systems of linear differential equations with randomly coupled, correlated coefficients. Using an asymptotically precise trace formula for analytic functions of random matrices and their adjoints, it rigorously derives and corrects the time decay rate in neural network models with asymmetric connectivity, generalizing prior results beyond independent coefficients.

ABSTRACT

We consider the long time asymptotic behavior of a large system of $N$ linear differential equations with random coefficients. We allow for general correlation structures c among the coefficients, thus we substantially generalize our previous work [14] that was restricted to the independent case. In particular, we analyze a recent model in the theory of neural networks [26] that specifically focused on the effect of the distributional asymmetry in the random connectivity matrix $X$. We rigorously prove and slightly correct the explicit formula from [27] on the time decay as a function of the asymmetry parameter. Our main tool is an asymptotically precise formula for the normalized trace of $f(X) g(X^*)$, in the large $N$ limit, where $f$ and $g$ are analytic functions.

Motivation & Objective

• To extend previous results on uncorrelated random coefficients to general correlation structures in large systems of linear differential equations.
• To rigorously analyze the time decay behavior in neural network models with asymmetric connectivity matrices, as studied in recent theoretical work.
• To correct and validate the explicit formula for decay rate as a function of asymmetry, previously proposed in [27].
• To develop a general asymptotic formula for the normalized trace of f(X)g(X*) in the large N limit, applicable to correlated random matrix ensembles.

Proposed method

• Derives an asymptotically precise formula for the normalized trace of f(X)g(X*) where X is a large random matrix with general correlation structure.
• Applies complex analysis techniques to analytic functions f and g, leveraging spectral properties of correlated random matrices.
• Uses the large N limit to approximate the dynamics of the system, focusing on the decay of solutions over time.
• Generalizes the trace formula beyond the independent coefficient case considered in prior work [14], allowing for arbitrary correlation structures.
• Validates and corrects the decay rate formula from [27] by deriving it from first principles using the trace asymptotics.

Experimental results

Research questions

• RQ1 How does the time decay of solutions to large systems of linear differential equations depend on the correlation structure of the random coefficients?
• RQ2 What is the precise asymptotic behavior of the normalized trace of f(X)g(X*) for correlated random matrices in the large N limit?
• RQ3 How does asymmetry in the connectivity matrix affect the long-time dynamics in neural network models?
• RQ4 Can the explicit decay rate formula proposed in [27] be rigorously derived and corrected using random matrix theory?

Key findings

• The paper derives an asymptotically precise formula for the normalized trace of f(X)g(X*) in the large N limit, valid for general correlation structures.
• It rigorously proves and slightly corrects the time decay rate formula from [27], showing that the decay depends explicitly on the asymmetry parameter of the connectivity matrix.
• The derived formula generalizes previous results restricted to independent coefficients, now accommodating arbitrary correlations.
• The method enables precise prediction of long-time behavior in neural network models with asymmetric random connectivity, resolving discrepancies in earlier heuristic derivations.